List of top Mathematics Questions on Tangents and Normals

Let $P$ be any point on the curve $x^{2/3}+y^{2/3}=a^{2/3}$. Then, what would be the length of the segment of the tangent between the coordinate axes?

If $x + 13y = 40$ is normal to the curve $y = 5x^{2} + \alpha x + \beta$ at the point (1,3), then the value of $\alpha\beta$ is equal to:

Find the equation of the normal to a parabola which is perpendicular to a given line. This involves:

For the curve $ y = 3x^3 - 3x^2 + 1 $ at $ x = 1 $, find the equation of the tangent.

The area of the triangle formed by the co-ordinate axes and a tangent to the curve $xy = a^2$ at the point $(x_1, y_1)$ is ______ sq. units (where $a, x_1$ and $y_1$ are non-zero)

A line passing through the point $ A(-2, 0) $, touches the parabola $ P: y^2 = x - 2 $ at the point $ B $ in the first quadrant. The area of the region bounded by the line $ AB $, parabola $ P $, and the x-axis is:

Find the equation of the tangent to the curve $ y = x^3 - 3x + 1 $ at the point where $ x = 2 $.

The value of $ \int \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx $ is equal to:

The portion of the line $ 4x + 5y = 20 $ in the first quadrant is trisected by the lines $ L_1 $ and $ L_2 $ passing through the origin. The tangent of an angle between the lines $ L_1 $ and $ L_2 $ is:

If $y = 4x - 5$ is a tangent to the curve $y^{2} = px^{3} + q$ at $(2, 3)$, then $p-q$ is

If the tangents at the points P and Q on the circle x² + y²– 2x + y = 5 meet at the point R $( \frac{9}{4},2 ) $then the area of the triangle PQR is

The number of common tangents, to the circles $ x^2 + y^2 - 18x - 15y + 131 = 0 $ and $ x^2 + y^2 - 6x - 6y - 7 = 0 $, is

The number of points on the curve $y=54 x^5-135 x^4-70 x^3+180 x^2+210 x$ at which the normal lines are parallel $to x+90 y+2=0$ is

Let a tangent to the curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4 x-6 y+11=0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$ Let $C_2$ be the image of $C_1$ in $T$ Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:

The tangents at the point $A (1,3)$ and $B (1,-1)$ on the parabola $y^2-2 x-2 y=1$ meet at the point $P$ Then the area (in unit $^2$ ) of $\triangle PAB$ is :-

Let $(\alpha, \beta, \gamma)$ $\text{ be the image of the point }$ $P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6$. $\text{ Then } \alpha + \beta + \gamma \text{ is equal to:}$

Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{ x^2}{36}+\frac{y^2}{4} =1$ meet the y-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line x = $2\sqrt{5}$ intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (α,β) , then α² - β² is equal to

The slope of tangent at any point (x, y) on a curve y = y(x) is $\frac{x^2+y^2}{2xy},x>0$ If y(2) = 0, then a value of y(8) is

Let the tangents at the points A(4, –11) and B(8, –5) on the circle $x^2+y^2-3x+10y-15=0$, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to