JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Circle

If the area of the triangle formed by the positive

x

-axis, the normal and the tangent to the circle

(x - 2)^{2} + (y - 3)^{2} = 25

at the point

(5, 7)

A

, then

24 A

is equal to ____

Answer: 1225

Solution

Center

C (2, 3)

, point

P (5, 7)

lies on the circle (radius

5

). Radius

C P

has slope

\frac{4}{3}

, so the tangent at

P

has slope

- \frac{3}{4}

3 x + 4 y = 43

, cutting the

x

-axis at

(\frac{43}{3}, 0)

. The normal is line

C P

4 x - 3 y + 1 = 0

, cutting the

x

-axis at

(- \frac{1}{4}, 0)

. The triangle has base between these intercepts,

\frac{43}{3} + \frac{1}{4} = \frac{175}{12}

, and apex

P

at height

7

A = \frac{1}{2} \cdot \frac{175}{12} \cdot 7 = \frac{1225}{24}

. Therefore

24 A = 1225

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Circle →

More JEE Main questions on Circle

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and $∠ AOB = π /3$ . Then the locus of the point of intersection of the lines PQ and MN is:
JEE Main 2026View solution →
If P is a point on the circle $x^{2} + y^{2} = 4$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.
JEE Main 2026View solution →
Let the set of all values of r, for which the circles $(x + 1)^{2} + (y + 4)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval ( $α$ , $β$ ). Then $α β$ is equal to
JEE Main 2026View solution →
Let a circle of radius 4 pass through the origin O , the points $A (- 3 a, 0)$ and $B (0, - 2 b)$ , where a and b are real parameters and $ab \neq = 0$ . Then the locus of the centroid of $△ OAB$ is a cir …
JEE Main 2026View solution →
Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord …
JEE Main 2026View solution →
Let the circle $x^{2} + y^{2} = 4$ intersect $x$ -axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α), 0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points …
JEE Main 2026View solution →