Assertion (A): Using the information in the given figure, we get CE : EA = 5:3.

Reason (R): Since, ∠ADE = ∠ABC = 90°

so, $\dfrac{CE}{EA} = \dfrac{BD}{AD} = \dfrac{8-5}{5} = \dfrac{3}{5}$

⇒ CE : EA = 3 : 5

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

A is false, R is true.

Explanation

Assertion (A) states that CE : EA = 5 : 3

Reason (R) states that CE : EA = 3 : 5

As these two ratios contradict one another, hence both Assertion (A) and Reason (R) can't be correct.

∴ We can rule out that Both A and R are true.

Analyzing the Reason (R):

In Δ ADE and Δ ABC,

∠ DAE = ∠ BAC (Common angle)

∠ ADE = ∠ ABC (Both are right angle)

∠ AED = ∠ ACB (Third angle of triangle will also be same)

Hence, Δ ADE ~ Δ ABC [By AA axiom of similarity]

$\therefore \dfrac{AE}{AC} = \dfrac{AD}{AB} = \dfrac{5}{8}$

Let AE = 5x and AC = 8x.

$\dfrac{CE}{EA} = \dfrac{CA - EA}{EA} \\[1em] \Rightarrow \dfrac{CE}{EA} = \dfrac{8x - 5x}{5x} \\[1em] \Rightarrow \dfrac{CE}{EA} = \dfrac{3x}{5x} \\[1em] \Rightarrow \dfrac{CE}{EA} = \dfrac{3}{5} \\[1em]$

∴ CE : EA = 3 : 5

∴ Reason (R) is true.

∴ Assertion (A) is false.

Hence, Assertion (A) is false, Reason (R) is true.